2D homogeneous solutions to the Euler equation

TL;DR

This paper classifies homogeneous solutions to the stationary Euler equation with finite energy, revealing trivial solutions for certain parameters and many new elliptic, hyperbolic, and parabolic solutions for others, with implications for Onsager's conjecture.

Contribution

The paper provides a comprehensive classification of homogeneous solutions to the Euler equation, identifying conditions for trivial and non-trivial solutions, and explores their structure and energy dissipation properties.

Findings

Only trivial solutions exist for 0<λ<1/2.

Many new elliptic, hyperbolic, and parabolic solutions are found for other λ values.

No anomalous energy dissipation occurs despite critical regularity.

Abstract

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show that only trivial solutions exist in the range $0<\lambda<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\lambda>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\lambda}) \cap \mathbb{N}$. The case $\lambda = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity 1/3.